Let me comment a bit on Chris Godsil's answer. The fact that $L_1\ne L_3$ follows from the easy-to-check fact that $L_1=\mathcal I\mathcal I^T$, where $\mathcal I$ is the incidence matrix of an arbitrary orientation of the graph. The nullity of $\mathcal I^T$ is the number of connected components of the graph, whereas the nullity of the transpose of the signless incidence matrix (which is what you presumably denote by $B^T$) is the number of *bipartite* connected components. So these matrices are clearly different. However, as pointed out by Chris Godsil, they yield the same information in the regular case: the difference is that the *upper end* of the spectrum of $L_3$ corresponds to the lower end of the spectrum of $L_1$, and vice versa: This was in fact, as far as I can judge, the main reasons for the introduction of $L_3$ in a 1994 paper by Desai and Rao.

The fact that the nullities of $L_1,L_2$ coincide can be seen by an argument that might be interesting for your purposes: $L_2$ (or at least the matrix $L_2:=D^{-\frac12}L_1 D^{-\frac12} $ suggested in the comment by Aaron Meyerowitz) is similar to $\tilde{L}_2:=D^{-1}L_1$, which is often called *normalized Laplacian* as well.
It turns out that both $L_1,\tilde{L}_2$ can be seen as Fréchet derivatives of the same energy functional $\mathcal E:f\mapsto\|\mathcal I^T f\|_{\ell^2(E)}^2$, but with respect to two different Hilbert spaces -- more precisely, to the vector space $\mathbb R^V$ endowed with the inner products
$$
(f|g):=\sum_{v\in V}f(v)g(v)
$$
and
$$
(f|g):=\sum_{v\in V}f(v)g(v)\deg(v)
$$
respectively (obvious modifications are needed in the case of infinite graphs).

normalizedLaplacian matrix and $L_1$ the standard Laplacian matrix. With certain exceptions (find them!) a normalized Laplacian is not a standard Laplacian. Read the article to see some common properties. $\endgroup$